Elliptic Weight Functions and Elliptic q-KZ Equation Hitoshi Konno Department of Mathematics, Tokyo University of Marine Science and Technology, Etchujima, Koto-ku, Tokyo 135-8533, Japan
[email protected] Abstract b N ), we present a systemBy using representation theory of the elliptic quantum group Uq,p (sl atic method of deriving the weight functions. The resultant slN type elliptic weight functions are new and give elliptic and dynamical analogues of those obtained in the rational and the trigonometric cases. We then discuss some basic properties of the elliptic weight functions. We also present an explicit formula for formal elliptic hypergeometric integral solution to the face type, i.e. dynamical, elliptic q-KZ equation.
1
Introduction
The weight functions are one of the main parts in hypergeometric integral solutions to the qKZ equations. See for example [44]. Recently new interests in the weight functions have been developing. Among others Gorbounov, Rim´anyi, Tarasov and Varchenko [16] have succeeded to identify the rational weight functions with the stable envelopes associated with the torus equivariant cohomology of the cotangent bundle to the partial flag variety. The stable envelopes were introduced by Maulik and Okounkov [33, 39] in more general setting associated with the equivariant cohomology of Nakajima’s quiver variety [36]. They are maps from the equivariant cohomology of the fixed point set of the torus action to the equivariant cohomology of the variety and play an important role in a formulation of a geometric representation theory of quantum groups on the equivariant cohomology. This identification has been extended to the equivariant K-theory case in [40] as well as to the dynamical version of the equivariant cohomology [41] and equivariant elliptic cohomology [14] cases both associated with the cotangent bundles to the Grassmannians. It has also been succeeded to construct a geometric representation of the Yangian Y (glN ) [16], the affine quantum b N ) [40], and the rational dynamical quantum group Ey (gl2 ) [41] and the elliptic group Uq (gl dynamical quantum group Eτ,y (gl2 ) [14] on the corresponding equivariant cohomology, K-theory, dynamical version of the cohomology and elliptic cohomology, respectively. There it is essential 1
to consider the finite-dimensional representations of the quantum groups on the Gelfand-Tsetlin basis of the tensor product of the vector representations. They are lifted to the geometric representations via the correspondence between the weight functions and the stable envelopes. It is also remarkable that the elliptic stable envelopes introduced by Aganagic and Okounkov [1] are the dynamical ones, where the K¨ahler variables play a role of the dynamical parameters. The purpose of this paper and subsequent papers is to extend these constructions to the b N ) [20, 25, 30], which is an elliptic and dynamical higher rank elliptic quantum group Uq,p (sl b N ) [4] and is isomorphic to the central extension analogue of the Drinfeld’s new realization of Uq (sl of Felder’s elliptic quantum group [10,30]. We expect that the elliptic weight functions of the slN type can be identified with the elliptic stable envelops [1] associated with the torus equivariant elliptic cohomology of the cotangent bundles to the partial flag variety. Such identification should b N ) on the equivariant elliptic cohomology. allow us to formulate a geometric action of Uq,p (sl b 2 case [14]. Recently, Felder, Rim´anyi and Varchenko has accomplished such study in the sl In this paper we discuss the elliptic weight functions of type slN and study their properties. We give a systematic derivation of the elliptic weight functions by using the vertex operators b N ) [20, 23, 24]. For the sl b 2 case we use the level-k representation of Uq,p (sl b 2 ), and the of Uq,p (sl resultant elliptic weight functions coincide with those obtained in [12, 44]. We discuss this case in a separate paper. We here concentrate on the higher rank level-1 representation and obtain a new result. The resultant elliptic weight functions are described by using the partitions of [1, n] in a combinatorial way and give elliptic and dynamical analogues to those obtained by Mimachi and Noumi [34, 35], Tarasov and Varchenko [40, 43]. The same method can be applied b N ) and allows us to derive to the trigonometric case too by using the vertex operators of Uq (sl b 2 case and in [34, 35, 40] for slN case. See the trigonometric weight functions in [32, 44] for sl also [18, 37, 43]. In the sequent paper we will discuss the finite-dimensional representations of b N ) on the Gelfand-Tsetlin basis and their geometric interpretations. Uq,p (sl Our derivation has the following advantages. 1) It makes a representation theoretical meaning of the combinatorial structure of the weight functions as well as of the partial flag variety transparent. Sec.4.1. 2) It makes the transition property of the weight function manifest. Sec.5.2. 3) It allows us to derive the shuffle algebra structure of the space of the weight functions. Sec.5.5 & AppendixC. As a byproduct we also give a new formula for formal elliptic hypergeometric integral solution to the face type elliptic q-KZ equation [11, 15]. 2
A part of the results has been presented in the workshops “Classical and Quantum integrable systems”, July 11-15, 2016, EIMI, St.Petersburg, “Recent Advances in Quantum Integrable Systems”, August 22-26, 2016, Univ.of Geneva and “Elliptic Hypergeometric Functions in Combinatorics, Integrable Systems and Physics”, March 20-24, 2017, ESI, Vienna. This paper is organized as follows. In Section 2 we prepare some notations including the elliptic dynamical R matrices. In Section 3 we review a construction of the vertex operators of b N )-modules. In particular we provide a free field realization of the vertex operators the Uq,p (sl for the level 1 representation. Section 4 is devoted to a derivation of the elliptic weight functions of the slN -type by using the vertex operators. In Section 5 we discuss some basic properties of the elliptic weight functions such as the triangular property, transition property, orthogonality, quasi-periodicity and the shuffle algebra structures. In Section 6, we give a formal elliptic hypergeometric integral solution to the face type elliptic q-KZ equation. In Appendix A we b N ). Appendix B is a summary summarize some basic facts on the elliptic quantum group Uq,p (sl b N ) both the finite and infinite-dimensional cases. In on the dynamical representations of Uq,p (sl Appendix C we provide a proof of Proposition 5.10 including a derivation of the shuffle algebra structure of the space of the weight functions. While preparing this manuscript, we become aware of the paper by Rim´anyi, V.Tarasov and A.Varchenko [42] which has some overlap with the content of the present paper.
2
Preliminaries
In this section we prepare the notation to be used in the text. Throughout this paper q are generic complex numbers satisfying |q| < 1 unless otherwise specified.
2.1
The commutative algebra H
b N = sl(N, b Let A = (aij ) (0 ≤ i, j ≤ N − 1) be the generalized Cartan matrix of sl C) [21]. Let N −1 ¯ ⊕ Cc, h ¯=⊕ b ¯ h=e h ⊕ Cd, e h=h i=1 Chi be the Cartan subalgebra of slN . Define δ, Λ0 , αi , Λi (1 ≤ i ≤ N − 1) ∈ h∗ by ¯ i , hj >= δi,j < αi , hj >= aji , < δ, d >= 1 =< Λ0 , c >, < Λ
(2.1)
¯∗ = ⊕N −1 CΛ ¯∗ ⊕ CΛ0 , Q = ⊕N −1 Zαi and P = ¯ i, e the other pairings are 0. We set h h∗ = h i=1 i=1 N −1 ¯ N ⊕i=1 ZΛi . Let {ϵj (1 ≤ j ≤ N )} be an orthonormal basis in R with the inner product ∑ (ϵj , ϵk ) = δj,k . We set ϵ¯j = ϵj − N k=1 ϵk /N . We realize the simple roots by αj = ϵj − ϵj+1 (1 ≤ ¯ j = ϵ¯1 + · · · + ϵ¯j (1 ≤ j ≤ N − 1). We define j ≤ N − 1) and the fundamental weights by Λ 3
¯ (1 ≤ j ≤ N ) by < ϵi , hϵ >= (ϵi , ϵj ) and hα ∈ h ¯ for α = ∑ cj ϵj , (cj ∈ C) by hϵj ∈ h j j ∑ ∗ ¯ ¯ hα = j cj hϵj . We regard h ⊕ h as the Heisenberg algebra by [hϵj , ϵk ] = (ϵj , ϵk ),
[hϵj , hϵk ] = 0 = [ϵj , ϵk ].
(2.2)
In particular, we have [hj , αk ] = ajk . We also set hj = hΛ¯ j . ¯∗ ) be the Heisenberg algebra defined by the commutation relations Let {Pα , Qβ } (α, β ∈ h [Pϵj , Qϵk ] = (ϵj , ϵk ), where Pα =
∑
j cj Pϵj
for α =
∑
j c j ϵj .
[Pϵj , Pϵk ] = 0 = [Qϵj , Qϵk ],
(2.3)
j N We set P¯h = ⊕N h = ⊕j=1 CQϵj Pj = Pαj , P = j=1 CPϵj , Q¯
PΛ¯ j and Qj = Qαj , Qj = QΛ¯ ∨ . j
−1 For the abelian group RQ = ⊕N i=1 ZQαi , we denote by C[RQ ] the group algebra over C of
RQ . We denote by eQα the element of C[RQ ] corresponding to Qα ∈ RQ . ∑ ∑ We define the commutative algebra H by H = e h ⊕ P¯h = j C(Pϵj + hϵj ) + j CPϵj + Cc. We denote the dual space of H by H ∗ = e h∗ ⊕ Q¯ . We define the paring by (2.1), < Qα , Pβ >= (α, β) h
and < Qα , hβ >=< Qα , c >=< Qα , d >= 0 =< α, Pβ >=< δ, Pβ >=< Λ0 , Pβ >. We often abbreviate Pϵj + hϵj as (P + h)ϵj and use (P + h)j,k = (P + h)ϵj − (P + h)ϵk , Pj,k = Pϵj − Pϵk , hj,k = hϵj − hϵk etc.. We define F = MH ∗ to be the field of meromorphic functions on H ∗ .
2.2
Infinite products
We use the following notations. [n]q =
q n − q −n , q − q −1
(x; q, t)∞ =
∞ ∏
(x; q)∞ =
∞ ∏
(1 − xq n ),
n=0
(1 − xq m tn ),
(x; p, q, t)∞ =
m,n=0
∞ ∏
(1 − xpl q m tn ),
l,m,n=0
Θp (z) = (z; p)∞ (p/z; p)∞ (p; p)∞ , (qt/x; q, t)∞ , Γ(x1 , x2 ; q, t) = Γ(x1 ; q, t)Γ(x2 ; q, t), Γ(x; q, t) = (x; q, t)∞ Γ(x; p, q, t) = (x; p, q, t)∞ (pqt/x; p, q, t)∞ . for |q| < 1, |t| < 1, |p| < 1.
2.3
Theta functions
Let r be a generic positive real number. We set p = q 2r . In general we consider the level k ∈ R representation of the elliptic quantum group Uq,p (g). See Appendix B. In that case we assume ∗
r∗ = r − k > 0 and set p∗ = q 2r . 4
We use the following Jacobi’s odd theta functions. [u] = q
u2 −u r
u2
[u]∗ = q r∗ −u Θp∗ (z),
Θp (z),
[u + r] = −[u],
(2.4)
[u + rτ ] = −e−πiτ e−2πiu/r [u], ∗
[u + r∗ ]∗ = −[u]∗ ,
(2.5) ∗
[u + r∗ τ ∗ ]∗ = −e−πiτ e−2πiu/r [u]∗ ,
∗
where z = q 2u , p = e−2πi/τ , p∗ = e−2πi/τ .
2.4
bN The elliptic dynamical R-matrix of type sl
b N ) given in Sec B.2 Let Vbz be the N -dimensional dynamical evaluation representation of Uq,p (sl and {vµ (µ = 1, · · · , N )} be its basis. We consider the following elliptic dynamical R-matrix e Vbz ) given by R+ (z1 /z2 , Π) ∈ EndC (Vbz ⊗ 1
2
R+ (z, Π) = ρ+ (z)R(z, Π), R(z, Π) =
N ∑
(2.6) ∑
Ej,j ⊗ Ej,j +
(
b(u, (P + h)j1 ,j2 )Ej1 ,j1 ⊗ Ej2 ,j2 + ¯b(z)Ej2 ,j2 ⊗ Ej1 ,j1
1≤j1 , ΦN (z1 ) · · · ΦN (zm ) =: ΦN (z1 ) · · · ΦN (zm ) : Fl (t1 ) · · · Fl (tn ) =: Fl (t1 ) · · · Fl (tn ) :
∏
1≤k .
1≤aSym , = (−)(N −1)/N zk Γ(q 2N zl /zk ; p, q 2N ) (q 2 tb /ta ; p)∞ (tb /ta ; p)∞ < Fl (ta )Fl (tb ) > = t2(r−1)/r a (ptb /ta ; p)∞ (pq −2 tb /ta ; p)∞ [va − vb − 1] = < Fl (ta )Fl (tb ) >Sym [va − vb ]
< ΦN (zk )ΦN (zl ) > =
11
with the symmetric parts (ta /tb ; p)∞ (tb /ta ; p)∞ , −2 a /tb ; p)∞ (pq tb /ta ; p)∞ (q 2 zk /zl , q 2 zl /zk ; p, q 2N )∞ < ΦN (zk )ΦN (zl ) >Sym = 2N . (q zk /zl , q 2N zl /zk ; p, q 2N )∞
< Fl (ta )Fl (tb ) >Sym = −(qta tb )1−1/r
(pq −2 t
(l)
(l)
4. For each l ∈ {1, · · · , N − 1}, symmetrize the integration variables t1 , · · · , tλ(l) . We denote this procedure by Symt(l) . Applying the above procedure to (4.2), we obtain the following. ϕµ1 ···µn (z1 , · · · , zn ) I I = dt1 · · · TN −µ1
I
(N −1)
TN −µn
(N −1)
×φµ1 (z1 , t1
dtn ΦN (z1 )FN −1 (t1 (µ1 )
, · · · , t1
; {Πµ1 ,m q −2 (N −1)
= TM
×
dt ΦN (z1 ) · · · ΦN (zn )FN −1 (t1 N −2 ∏ λ(l) ∏
(l+1) λ∏
l=1 a=1
b=1 (l) (l+1) ia k=2
1≤k
1≤k k=2 il+1.λl+1 then nj=s+1 δµj ,l+1 = 0.
4.3
Entire function version
Let us set Hλ (t, z) :=
(l+1) [ N −1 ∏ λ(l) λ∏ ∏
(l+1)
vb
l=1 a=1 b=1
− va(l) +
] 1 . 2
(4.7)
fI The following gives an entire function version of W fI (t, z, Π) = Sym (1) · · · Sym (N −1) UI (t, z, Π), WI (t, z, Π) = Hλ (t, z)W t t where UI (t, z, Π) =
[ ] (l+1) (l) 1 v − v + (P + h) − C (s) − a µ ,l+1 µ ,l+1 s s 2 [1] b [(P + h)µs ,l+1 − Cµs ,l+1 (s)] (N ) a=1
N −1 ∏ λ(l) ∏ l=1
∏
λ(l+1)
×
(4.8)
b=1 (l+1) (l) i >ia b
is
[ (l+1)
vb
− va(l) −
1 2
]
∏
λ(l+1)
i
b=1 (l+1) (l) ib
−
va(l)
is located in the lower row than
] 1 − . 2 (l+1)
) (l = 1, · · · , N − 1), where tb
is located in the upper row than
), assign [ (l+1) vb
(l)
(N )
= is ).
), assign [
(l)
,
−
va(l)
] 1 + . 2
(l)
• for each pair (ta , tb ) (1 ≤ a < b ≤ λ(l) ) in the l-th column (l = 1, · · · , N − 1), assign (l)
(l)
[vb − va + 1] (l)
(l)
[vb − va ]
.
Example 3. For the case N = 3, n = 4, I = {I1 = {2}, I2 = {1, 4}, I3 = {3}} = I2132
(1) t1
t1
(2)
z1
(2) t2
z2 z3
(2) t3
15
z4
we obtain UI (t, z, Π) (2)
=
(1)
(2)
[v2 − v1 + (P + h)1,2 − C1,2 (2) − 1/2][1] [u1 − v1 + (P + h)2,3 − C2,3 (1) − 1/2][1] [(P + h)1,2 − C1,2 (2)] [(P + h)2,3 − C2,3 (1)] (2)
(2)
[u2 − v2 + (P + h)1,3 − C1,3 (2) − 1/2][1] [u4 − v3 + (P + h)2,3 − C2,3 (4) − 1/2][1] × [(P + h)1,3 − C1,3 (2)] [(P + h)2,3 − C2,3 (4)] (2)
(1)
(2)
(2)
(2)
(2)
(2)
(1)
(2)
(2)
(2)
(2)
(2)
×[v3 − v1 − 1/2][u2 − v1 − 1/2][u3 − v1 − 1/2][u4 − v1 − 1/2][u3 − v2 − 1/2][u4 − v2 − 1/2] ×[v1 − v1 + 1/2][u1 − v2 + 1/2][u1 − v3 + 1/2][u2 − v3 + 1/2][u3 − v3 + 1/2] ∏ [v (2) − va(2) + 1] b . (2) (2) [v − v ] a 1≤a .
j=3,4
For C2,3 (1), we have s = i2,1 = 1 < i3,1 = 3, m2,3 (1) = 1. Hence ∑
C2,3 (1) = 2 − 1 − 1 + 1 − 1 = 0 =
< ϵ¯µj , h2,3 > .
j=2,3,4
For C1,3 (2), we have s = i1,1 = 2 < i3,1 = 3, m1,3 (2) = 1. Hence C1,3 (2) = 1 − 1 − 1 + 1 − 1 = −1 =
∑ j=3,4
For C2,3 (4), we have s = i2,2 = 4 > i3,1 = 3. Hence C2,3 (4) = 2 − 2 = 0.
16
< ϵ¯µj , h1,3 > .
5
Properties of the Elliptic Weight Functions
In this section we discuss some basic properties of the elliptic weight functions. We consider (l)
the following weight functions obtained from those in the last section by va
(l)
7→ va + l/2
(l = 1, · · · , N, a = 1, · · · , λ(l) ). WI (t, z, Π) = Symt(1) · · · Symt(N −1) UI (t, z, Π), [ ] (l+1) (l) N −1 ∏ λ(l) v − v + (P + h) − C (s) [1] ∏ a µs ,l+1 µs ,l+1 b UI (t, z, Π) = [(P + h)µs ,l+1 − Cµs ,l+1 (s)] l=1 a=1
[ ] (l+1) vb − va(l)
∏
λ(l+1)
×
b=1 (l+1) (l) i >ia b
∏
λ(l+1)
(5.1)
(N )
is
(l+1)
=ib
(l)
=ia
] ∏ [v (l) − v (l) + 1] [ a (l+1) b . vb − va(l) + 1 (l) (l) [v − v ] a b=a+1 b λ(l)
b=1 (l+1) (l) i ), [d, g(P + h)] = [d, g(P )] = 0,
(A.6)
∂ ∂ ej (z), [d, fj (z)] = −z fj (z), ∂z ∂z Ki± ej (z) = q ∓aij ej (z)Ki± , Ki± fj (z) = q ±aij fj (z)Ki± , [aij m]q [cm]q 1 − pm −cm [αi,m , αj,n ] = δm+n,0 q , m 1 − p∗m [aij m]q 1 − pm −cm m [αi,m , ej (z)] = q z ej (z), m 1 − p∗m [aij m]q m [αi,m , fj (z)] = − z fj (z), m [d, αj,n ] = nαj,n ,
(A.5)
[d, ej (z)] = −z
28
(A.7) (A.8) (A.9) (A.10) (A.11)
(q aij z2 /z1 ; p∗ )∞ (q aij z1 /z2 ; p∗ )∞ e (z )e (z ) = −z ej (z2 )ei (z1 ), i 1 j 2 2 −a ∗ ∗ (p q ij z2 /z1 ; p )∞ (p∗ q −aij z1 /z2 ; p∗ )∞ (q −aij z2 /z1 ; p)∞ (q −aij z1 /z2 ; p)∞ z1 f (z )f (z ) = −z fj (z2 )fi (z1 ), i 1 j 2 2 (pq aij z2 /z1 ; p)∞ (pq aij z1 /z2 ; p)∞ ) δi,j ( −c − 2c + − 2c c [ei (z1 ), fj (z2 )] = δ(q z /z )ψ (q z ) − δ(q z /z )ψ (q z ) , 1 2 j 2 1 2 j 2 q − q −1 z1
(A.12) (A.13) (A.14)
{ (p∗ q 2 z2 /z1 ; p∗ )∞ (p∗ q −1 z/z1 ; p∗ )∞ (p∗ q −1 z/z2 ; p∗ )∞ ei (z1 )ei (z2 )ej (z) (p∗ q −2 z2 /z1 ; p∗ )∞ (p∗ qz/z1 ; p∗ )∞ (p∗ qz/z2 ; p∗ )∞ (p∗ q −1 z/z1 ; p∗ )∞ (p∗ q −1 z2 /z; p∗ )∞ − [2]q ei (z1 )ej (z)ei (z2 ) (p∗ qz/z1 ; p∗ )∞ (p∗ qz2 /z; p∗ )∞ } (p∗ q −1 z1 /z; p∗ )∞ (p∗ q −1 z2 /z; p∗ )∞ + ej (z)ei (z1 )ei (z2 ) + (z1 ↔ z2 ) = 0, (p∗ qz1 /z; p∗ )∞ (p∗ qz2 /z; p∗ )∞ (A.15) { −2 (pqz/z1 ; p)∞ (pqz/z2 ; p)∞ (pq z2 /z1 ; p)∞ fi (z1 )fi (z2 )fj (z) 2 (pq z2 /z1 ; p)∞ (pq −1 z/z1 ; p)∞ (pq −1 z/z2 ; p)∞ (pqz/z1 ; p)∞ (pqz2 /z; p)∞ − [2]q fi (z1 )fj (z)fi (z2 ) (pq −1 z/z1 ; p)∞ (pq −1 z2 /z; p)∞ } (pqz1 /z; p)∞ (pqz2 /z; p)∞ + fj (z)fi (z1 )fi (z2 ) + (z1 ↔ z2 ) = 0, (pq −1 z1 /z; p)∞ (pq −1 z2 /z; p)∞ for |i − j| = 1, (A.16) where p∗ = pq −2c and δ(z) =
∑
n∈Z z
n.
We treat these relations as formal Laurent series in z, w and zj ’s. All the coefficients are well-defined in the p-adic topology.
A.2
The orthonormal basis type elliptic bosons
j Let us define the orthonormal basis type elliptic bosons Em (1 ≤ j ≤ N, m ∈ Z̸=0 ) [6] by j−1 N −1 ∑ ∑ Cm −N m j Em = q jm (A.17) −q [km]q αk,m + [(N − k)m]q αk,m q − q −1 k=1
k=j
with Cm =
1 [m]2q [N m]q
One can show the following relations.
29
.
Proposition A.2. N ∑
j q (j−1)m Em = 0,
(A.18)
j=1 j , Enj ] = δm+n,0 [Em
[cm]q [ηm]q [(N − 1)m]q 1 − pm −cm q , m(q − q −1 )2 [m]3q [N m]q 1 − p∗m
j [Em , Enk ] = −δm+n,0 q (sgn(k−j)N −k+j)m
where
[cm]q 1 − pm −cm q , m(q m − q −m )2 [N m]q 1 − p∗m
(A.19) (A.20)
+ (l > j) sgn(l − j) = 0 (l = j) − (l < j).
Note also Proposition A.3. j j+1 αj,m = [m]2q (q − q −1 )(Em − q −m Em ).
(A.21)
Furthermore if we set Ajm
= (q − q m
−m
)
j ∑
k q (k−j−1)m Em ,
k=1
then we have [αi,m , Ajm ] = −δi,j δm+n,0
[cm]q 1 − pm −cm q m 1 − p∗m
(1 ≤ i, j ≤ N − 1)
Hence we call Ajm the fundamental weight type bosons.
A.3
The elliptic currents kj (z)
Let us set
∑ αj,m m −m ψj (z) =: exp (q − q −1 ) p z :. 1 − pm
(A.22)
m̸=0
Here :
: denotes the normal ordering defined by α α j,m k,n if m ≤ n : αj,m αk,n := αk,n αj,m if m > n
for 1 ≤ j, k ≤ N − 1. Then the elliptic currents ψj± (z) in Definition A.1 can be written as ψj− (q − 2 z) = Kj− ψj (pq −c z).
ψj+ (q − 2 z) = Kj+ ψj (z),
c
c
30
(A.23)
j Let us introduce the new elliptic currents kj (z) (j ∈ I ∪ {N }) associated with Em by ∑ [m]2 (q − q −1 )2 q m j −m kj (z) = : exp p E z : m 1 − pm
(A.24)
m̸=0
Then from Proposition A.3 we obtain the following relations. Proposition A.4. ψj (z) = ϱkj (z)kj+1 (qz)−1 ,
(A.25)
where ϱ=
(p; p)∞ (p∗ q 2 ; p∗ )∞ . (p∗ ; p∗ )∞ (pq 2 ; p)∞
(A.26)
In addition, from Proposition A.2 we obtain the following commutation relations. Theorem A.5. ρ˜+∗ (z) kj (z2 )kj (z1 ), (1 ≤ j ≤ N ), ρ˜+ (z) ρ˜+∗ (z) Θp∗ (q −2 z)Θp (z) kk (q k z2 )kj (q j z1 ) kj (q j z1 )kk (q k z2 ) = + ρ˜ (z) Θp∗ (z)Θp (q −2 z)
kj (z1 )kj (z2 ) =
(1 ≤ j < k ≤ N ),
where z = z1 /z2 , ρ˜+ (z) is given in (2.8) and ρ+∗ (z) = ρ+ (z)|p7→p∗ . Proposition A.6. Θp∗ (q −c z) ej (z2 )kj (z1 ) (1 ≤ j ≤ N ), Θp∗ (q −c−2 z) Θp∗ (q −c−1 z) kj (z1 )ej−1 (z2 ) = ej−1 (z2 )kj (z1 ) (2 ≤ j ≤ N ), Θp∗ (q −c+1 z) kj (z1 )ek (z2 ) = ek (z2 )kj (z1 ) (k ̸= j, j − 1), −2 Θp (q z) kj (z1 )fj (z2 ) = fj (z2 )kj (z1 ) (1 ≤ j ≤ N ), Θp (z) Θp (qz) kj (z1 )fj−1 (z2 ) = fj−1 (z2 )kj (z1 ) (2 ≤ j ≤ N ), Θp (q −1 z) kj (z1 )fk (z2 ) = fk (z2 )kj (z1 ) (k ̸= j, j − 1). kj (z1 )ej (z2 ) =
A.4
Modified elliptic currents (1)
The R matrix (2.7) is gauge equivalent to Jimbo-Miwa-Okado’s AN −1 face type Boltzmann weight [17] and conveniently expressed by using Jacobi’s theta function (2.4). However one drawback is that Jacobi’s theta is accompanied by the fractional power of z. In order to deal with this one needs to introduce the following modifications of the elliptic currents [23]. 31
Definition A.7. We introduce the new generators e±ζϵ¯j (1 ≤ j ≤ N ) satisfying eQϵ¯j eQϵ¯k = q ( r − r∗ )sgn(j−k) eQϵ¯k eQϵ¯j , 1
eQϵ¯j eζϵ¯k = q
1
1 sgn(j−k) r
(A.27)
eζϵ¯k eQϵ¯j ,
(A.28)
1
eζϵ¯j eζϵ¯k = q r sgn(j−k) eζϵ¯k eζϵ¯j , e±
[Pϵ¯j , e±ζϵ¯k ] = 0,
(A.29)
∑N
¯j j=1 ζϵ
= 1,
(A.30)
b N )] = 0. [e±ζϵ¯j , C[Q]] = [e±ζϵ¯j , Uq,p (sl
(A.31)
Let us set ζj = ζϵ¯j − ζϵ¯j+1 , we have Proposition A.8. eQαj eQαk = q ( r − r∗ )(δj,k+1 −δj,k−1 ) eQαk eQαj , 1
eQϵ¯j eQαk = q (
1
1 − r1∗ r
(A.32)
)(δj,k +δj,k+1 ) eQαk eQϵ¯j ,
(A.33)
1
eQϵ¯j eζk = q r (δj,k +δj,k+1 ) eζk eQϵ¯j ,
(A.34)
eQαj eζk = q (δj,k+1 −δj,k−1 ) eζk eQαj ,
(A.35)
eζj eζk = q r (δj,k+1 −δj,k−1 ) eζk eζj ,
(A.36)
b N )] = 0. [Pϵ¯k , e±ζj ] = [e±ζj , Uq,p (sl
(A.37)
1 r
1
Definition A.9. We define the modified elliptic currents as follows. Ej (z) = ej (z)eζj (q N −j z)− Fj (z) = fj (z)e−ζj (q N −j z)
Pαj −1 r∗
(1 ≤ j ≤ N − 1),
(P +h)αj −1
(1 ≤ j ≤ N − 1),
r
Kj+ (z) = k+j (q j−N +1 z)e−Qϵ¯j q −hϵ¯j (q −r+1 z)− r∗ (Pϵ¯j −1)− r ((P +h)ϵ¯j −1) , ( ) Kj− (z) = Kj+ pq −c z . 1
1
for 1 ≤ j ≤ N . We also set ( ) ( )−1 c c ± Hj± (z) = ϱKj± q N −j−1 q 2 z Kj+1 q N −j−1 q 2 z
(A.38)
= ψj± (z)(Kj± )−1 e−Qαj q ∓hj (q N −j q ± 2 z)− r∗ (Pαj −1)+ r ((P +h)αj −1) , 1
c
1
(1 ≤ j ≤ N − 1).
and N N 1 ∑ 1 ∑ db = d + ∗ (Pj + 2)P j − ((P + h)j + 2)(P + h)j . 2r 2r j=1
(A.39)
j=1
b N ) can be rewritten as follows in the sense Then the defining relations (A.7)–(A.16) of Uq,p (sl of analytic continuation. 32
Proposition A.10. [hi , αj,n ] = 0,
[hi , Ej (z)] = aij Ej (z),
[hi , Fj (z)] = −aij Fj (z),
b hi ] = 0, [d,
b αi,n ] = nαi,n , [d, ) ( ) ( 1 ∂ 1 ∂ b b + Ei (z), [d, Fi (z)] = −z + Fi (z), [d, Ei (z)] = −z ∂z r∗ ∂z r [aij m]q [cm]q 1 − pm −cm [αi,m , αj,n ] = δm+n,0 q , m 1 − p∗m [aij m]q 1 − pm −cm m [αi,m , Ej (z)] = q z Ej (z), m 1 − p∗m [aij m]q m z Fj (z), [αi,m , Fj (z)] = − m [ [ aij ]∗ aij ]∗ u−v− Ei (z)Ej (w) = u − v + Ej (w)Ei (z), 2 ] 2] [ [ aij aij u−v+ Fi (z)Fj (v) = u − v − Fj (w)Fi (z), 2 2 ( c z ) + −c/2 ) δi,j ( ( −c z ) − c/2 [Ei (z), Fj (w)] = H (q w) − δ q H (q w) , δ q i q − q −1 w w i
(A.40) (A.41) (A.42) (A.43) (A.44) (A.45)
(A.46) (A.47) (A.48)
{ ∗ −1 z/z ; p∗ ) (p∗ q −1 z/z ; p∗ ) 1 (p q (p∗ q 2 z2 /z1 ; p∗ )∞ 1 ∞ 2 ∞ r∗ (z /z) Ei (z1 )Ei (z2 )Ej (z) 2 ∗ −2 ∗ ∗ ∗ ∗ ∗ (p q z2 /z1 ; p )∞ (p qz/z1 ; p )∞ (p qz/z2 ; p )∞ (p∗ q −1 z/z1 ; p∗ )∞ (p∗ q −1 z2 /z; p∗ )∞ Ei (z1 )Ej (z)Ei (z2 ) − [2]q (p∗ qz/z1 ; p∗ )∞ (p∗ qz2 /z; p∗ )∞ } ∗ −1 z /z; p∗ ) (p∗ q −1 z /z; p∗ ) 1 (p q 1 ∞ 2 ∞ ∗ + (z/z1 ) r Ej (z)Ei (z1 )Ei (z2 ) + (z1 ↔ z2 ) = 0, (p∗ qz1 /z; p∗ )∞ (p∗ qz2 /z; p∗ )∞ (A.49) { 1 (pq −2 z /z ; p) 1 (pqz/z1 ; p)∞ (pqz/z2 ; p)∞ 2 1 ∞ (z/z2 ) r Fi (z1 )Fi (z2 )Fj (z) z1r (pq 2 z2 /z1 ; p)∞ (pq −1 z/z1 ; p)∞ (pq −1 z/z2 ; p)∞ (pqz/z1 ; p)∞ (pqz2 /z; p)∞ − [2]q Fi (z1 )Fj (z)Fi (z2 ) (pq −1 z/z1 ; p)∞ (pq −1 z2 /z; p)∞ } 1 (pqz1 /z; p)∞ (pqz2 /z; p)∞ + (z1 /z) r F (z)F (z )F (z ) + (z1 ↔ z2 ) = 0 (|i − j| = 1). j i 1 i 2 (pq −1 z1 /z; p)∞ (pq −1 z2 /z; p)∞ (A.50) − 1∗ z1 r
In addition, one can rewrite the formulas in Theorem A.5 and Proposition A.6 as follows.
33
Proposition A.11. ρ+∗ (z1 /z2 ) + Kj (z2 )Kj+ (z1 ), + ρ (z1 /z2 ) +∗ (z /z ) [u − u − 1]∗ [u − u ] ρ 1 2 1 2 1 2 Kj+ (z1 )Kl+ (z2 ) = + K + (z2 )Kj+ (z1 ) (1 ≤ j < l ≤ N ), ρ (z1 /z2 ) [u1 − u2 ]∗ [u1 − u2 − 1] l [ ]∗ u1 − u2 + j−N2−c+1 ]∗ Ej (z2 )Kj+ (z1 ) Kj+ (z1 )Ej (z2 ) = [ (1 ≤ j ≤ N ), j−N −c+1 u1 − u2 + − 1 2 [ ]∗ u1 − u2 + j−N2+1−c + + ]∗ Ej (z2 )Kj+1 Kj+1 (z1 )Ej (z2 ) = [ (z1 ) (1 ≤ j ≤ N − 1), j−N +1−c u1 − u2 + + 1 2
Kj+ (z1 )Kj+ (z2 ) =
Kl+ (z1 )Ej (z2 ) = Ej (z2 )Kl+ (z1 ) (l ̸= j, j + 1), ] [ u1 − u2 + j−N2 +1 − 1 ] Fj (z2 )Kj+ (z1 ) (1 ≤ j ≤ N ), Kj+ (z1 )Fj (z2 ) = [ j−N +1 u1 − u2 + 2 [ ] u1 − u2 + j−N2 +1 + 1 + + ] Fj (z2 )Kj+1 Kj+1 (z1 )Fj (z2 ) = [ (z1 ) (1 ≤ j ≤ N − 1), j−N +1 u1 − u2 + 2 (l ̸= j, j + 1).
Kl+ (z1 )Fj (z2 ) = Fj (z2 )Kl+ (z1 )
A.5
The half currents and the L-operators
b N ) as follows. We define the half currents of Uq,p (sl Definition A.12. [23] Let us assume |p| < |z| < 1. We set I l−1 ∏ dtm + Fl−1 (tl−1 )Fl−2 (tl−2 ) · · · Fj (tj ) Fj,l (z) = aj,l 2πitm Tl−j m=j
×
l−2 − 1][1] ∏ [vm+1 − vm + (P + h)j,m+1 − 21 ][1] ,(A.51) [vm+1 − vm + 21 ][Pj,m+1 + hj,m+1 ] [u − vl−1 + l−N 2 ][Pj,l + hj,l − 1] m=j
[u − vl−1 + (P + h)j,l +
+ El,j (z)
=
×
a∗j,l
I
l−N 2
l−1 ∏ dtm El−1 (tl−1 )El−2 (tl−2 ) · · · Ej (tj ) 2πitm
Tl−j m=j
l−2 l−N c ∗ ∗ ∏ 2 − 2 + 1] [1] l−N c ∗ ∗ 2 − 2 ] [Pj,l − 1] m=j
[u − vl−1 − Pj,l + [u − vl−1 +
[vm+1 − vm − Pj,m+1 + 21 ]∗ [1]∗ , [vm+1 − vm − 21 ]∗ [Pj,m+1 − 1]∗
(A.52)
where z = q 2u , ta = q 2va (a = j, j + 1, · · · , l − 1) and Tl−j = {t ∈ Cl−j | |tj | = · · · = |tl−1 | = 1}. The constants aj,l and a∗j,l are chosen to satisfy −
ϱ aj,l a∗j,l [1] = 1. q − q −1 [0]′ 34
(A.53)
+ + We call Fj,l (z), El,j (z), (1 ≤ j < l ≤ N ) and Kj+ (z) (j = 1, · · · , N ) the half currents.
b + (z) ∈ End(CN ) ⊗ Definition A.13. By using the half currents, we define the L-operator L b N ) as follows. Uq,p (sl b + (z) = L
+ + (z) · · · 1 F1,2 (z) F1,3
0 .. . .. . 0
+ F2,3 (z) · · · .. .. . . .. . 1
1 ..
.
···
···
0
K + (z) 0 ··· 1 + (z) F2,N 0 K2+ (z) .. . .. .. . . + FN −1,N (z) 0 ··· 0 1 + F1,N (z)
1
··· .. . .. . .. .
0
+ E2,1 (z) 1 + + × E3,1 (z) E3,2 (z) .. .. . . + + EN,1 (z) EN,2 (z) · · ·
···
0 .. . .. .
0 .. .
0 + KN (z)
.. . . 1 0 + EN,N (z) 1 −1
(A.54)
We conjecture that the L-operator satisfies the following dynamical RLL-relation [19]. Conjecture A.14. [23] b +(1) (z1 )L b +(2) (z2 ) = L b +(2) (z2 )L b +(1) (z1 )R+∗(12) (z1 /z2 , Π∗ ). R+(12) (z1 /z2 , Π)L
A.6
(A.55)
bN ) The H-Hopf algebroid Uq,p (sl
Let A be a complex associative algebra, H be a finite dimensional commutative subalgebra of A, and MH∗ be the field of meromorphic functions on H∗ the dual space of H. Definition A.15 (H-algebra [5]). An H-algebra is an associative algebra A with 1, which is ⊕ bigraded over H∗ , A = Aα,β , and equipped with two algebra embeddings µl , µr : MH∗ → α,β∈H∗
A0,0 (the left and right moment maps), such that µl (fb)a = aµl (Tα fb),
µr (fb)a = aµr (Tβ fb),
a ∈ Aα,β , fb ∈ MH∗ ,
where Tα denotes the automorphism (Tα fb)(λ) = fb(λ + α) of MH∗ . e is the H∗ -bigraded vector space Let A and B be two H-algebras. The tensor product A⊗B with e αβ = (A⊗B)
⊕ γ∈H∗
(Aαγ ⊗MH∗ Bγβ ), 35
where ⊗MH∗ denotes the usual tensor product modulo the following relation. B b b µA r (f )a ⊗ b = a ⊗ µl (f )b,
a ∈ A, b ∈ B, fb ∈ MH∗ .
(A.56)
e is again an H-algebra with the multiplication (a ⊗ b)(c ⊗ d) = ac ⊗ bd The tensor product A⊗B and the moment maps e
e
⊗B µA = µA l ⊗ 1, l
⊗B µA = 1 ⊗ µB r r.
b N ) is an H-algebra by Proposition A.16. [28, 30] U = Uq,p (sl U=
⊕
Uα,β
α,β∈H ∗
Uα,β
{ = x ∈ U q P +h xq −(P +h) = q x,
P
q xq
−P
=q
x ∀P + h, P ∈ H
}
and µl , µr : F → U0,0 defined by µl (fb) = f (P + h, p) ∈ F[[p]],
µr (fb) = f (P, p∗ ) ∈ F[[p]].
We regard Tα = eα ∈ C[RQ ] as the shift operator MH ∗ → MH ∗ (Tα fb) = eα f (P, p∗ )e−α = f (P + < α, P >, p∗ ). Hereafter we abbreviate f (P + h, p) and f (P, p∗ ) as f (P + h) and f ∗ (P ), respectively. We also consider the H-algebra of the shift operators D={
∑
fbα Tα | fbα ∈ MH ∗ , α ∈ RQ },
α
Dα,α = { fbT−α },
Dα,β = 0 (α ̸= β),
D b b b µD l (f ) = µr (f ) = f T0
fb ∈ MH ∗ .
Then we have the H-algebra isomorphism e ∼ e U∼ = U ⊗D = D⊗U.
(A.57)
We define two H-algebra homomorphisms, the co-unit ε : U → D and the co-multiplication
36
e as well as the algebra antihomomorphism S : U → U by ∆ : U → U ⊗U b + (z)) = δi,j TQϵ ε(L i,j i
(n ∈ Z),
ε(eQ ) = eQ ,
(A.58)
ε(µl (fb)) = ε(µr (fb)) = fbT0 , ∑ b + (z)) = b + (z)⊗ b + (z), eL ∆(L L i,j i,k k,j
(A.59) (A.60)
k Q
e , ∆(eQ ) = eQ ⊗e e ∆(µl (fb)) = µl (fb)⊗1,
(A.61) e r (fb), ∆(µr (fb)) = 1⊗µ
(A.62)
b + (z)) = (L b + (z)−1 )ij , S(L ij S(eQ ) = e−Q ,
S(µr (fˆ)) = µl (fˆ),
(A.63) S(µl (fˆ)) = µr (fˆ).
(A.64)
b N ), H, MH ∗ , µl , µr , ∆, ε, S) becomes an H-Hopf algebroid [5, 22, 28, 30]. Then the set (Uq,p (sl
B B.1
Representations Dynamical representations
Let us consider a vector space Vb over F = MH ∗ , which is H-diagonalizable, i.e. ⊕ Vb = Vbλ,ν , Vbλ,ν = {v ∈ Vb | q P +h · v = q v, q P · v = q v ∀P + h, P ∈ H}. λ,ν∈H ∗
Let us define the H-algebra DH,Vb of the C-linear operators on Vb by ⊕ DH,Vb = (DH,Vb )α,β , α,β∈H ∗
f (P + h)X = Xf (P + h+ < α, P + h >), X ∈ EndC Vb , = f (P )X = Xf (P + < β, P >) f (P ), f (P + h) ∈ F, X · Vbλ,µ ⊆ Vbλ+α,µ+β
(DH,Vb )α,β
DH,Vb
µl
(fb)v = f (< λ, P + h >, p)v,
D b µr H,V (fb)v = f (< ν, P >, p∗ )v,
fb ∈ F, v ∈ Vbλ,ν .
b N ) on Vb to be an Definition B.1. [5, 22, 28] We define a dynamical representation of Uq,p (sl b N ) → D b . By the action of Uq,p (sl b N ) we regard Vb as a H-algebra homomorphism π : Uq,p (sl H,V b N )-module. Uq,p (sl b N )-module has level k if c act as the scalar k Definition B.2. For k ∈ C, we say that a Uq,p (sl on it. b N ) generated by c, d, K ± (i ∈ I), Definition B.3. Let H, N+ , N− be the subalgebras of Uq,p (sl i by αi,n (i ∈ I, n ∈ Z>0 ), ei,n (i ∈ I, n ∈ Z≥0 ) fi,n (i ∈ I, n ∈ Z>0 ) and by αi,−n (i ∈ I, n ∈ Z>0 ), ei,−n (i ∈ I, n ∈ Z>0 ), fi,−n (i ∈ I, n ∈ Z≥0 ), respectively. 37
b N )-module Vb (λ, ν) is Definition B.4. For k ∈ C, λ ∈ h∗ and ν ∈ H ∗ , a (dynamical) Uq,p (sl called the level-k highest weight module with the highest weight (λ, ν), if there exists a vector v ∈ Vb (λ, ν) such that b N ) · v, Vb (λ, ν) = Uq,p (sl c · v = kv,
B.2
N+ · v = 0,
f (P ) · v = f (< ν, P >)v,
f (P + h) · v = f (< λ, P + h >)v.
The N -dimensional dynamical evaluation representation
Let Vb =
N ⊕
Fvµ ⊗ 1 and set Vbz = Vb [z, z −1 ]. Let eQα ∈ C[RQ ] act on f (Pβ )v⊗1 by eQα (f (Pβ )v⊗
µ=1
1) = f (Pβ − (α, β))v ⊗ 1. Theorem B.5. Let Ej,k (1 ≤ j, k ≤ N ) denote the matrix units such that Ej,k vµ = δk,µ vj . The b N ) on Vbz . following gives the N -dimensional dynamical evaluation representation of Uq,p (sl πz (c) = 0,
πz (d) = −z
d , dz
[m]q j−N +1 n −m (q z) (q Ej,j − q m Ej+1,j+1 ), m ( ) (pq 2 ; p)∞ πz (ej (w)) = Ej,j+1 δ q j−N +1 z/w e−Qαj , (p; p)∞ ( ) (pq −2 ; p)∞ πz (fj (w)) = Ej+1,j δ q j−N +1 z/w , (p; p)∞ πz (αj,m ) =
πz (ψj+ (w, p)) = q −π(hj ) e−Qαj πz (ψj− (w, p)) = q π(hj ) e−Qαj
Θp (q −j+N −1+2π(hj ) wz ) Θp (q −j+N −1 w/z)
(1 ≤ j ≤ N − 1),
Θp (q j−N +1−2π(hj ) wz ) . Θp (q j−N +1 z/w)
Here π(hj ) = Ej,j − Ej+1,j+1 . Combining the formulas in Definition A.9, A.12, A.13 and Theorem B.5, we obtain Corollary B.6. b + (w))k,l = R+ (w/z, Π∗ )jl . πz (L i,j ik
B.3
The level-1 representation
b N ). We mainly follow the work [6]. Next we consider level-1 (c = 1) representation of Uq,p (sl It is convenient to extend the root lattice Q by adding the elements ζj (j = 1, · · · , N − 1) in b = ⊕j Zb Definition A.7. Let us set α bj = αj + ζj and consider Q αj . We define the extended group b with assuming the following central extension. algebra C[Q] eαi eαj = (−1)(αi ,αj ) eαj eαi . 38
∑ b 0 = Λ0 . ∈ h∗ , we also set ζω = j cj ζϵ¯j and ω b = ω + ζω . Set also Λ b N . For generic ¯ a (a = 1, · · · , N − 1) be the fundamental weights of sl Let Λ0 and Λa = Λ0 + Λ For ω =
∑
¯j j cj ϵ
ν ∈ h∗ , we set b b ⊗ eQν¯ C[RQ ], Vb (Λa + ν, ν) = F ⊗C (Fα,1 ⊗ eΛa C[Q])
where Fα,1 = C[{αj,−m (j = 1, · · · , N − 1, m ∈ N>0 )}]. Then we have the following decomposition. ⊕
Vb (Λa + ν, ν) =
Fa,ν (ξ, η),
ξ,η∈Q
where b
b
Fa,ν (ξ, η) = F ⊗C (Fα,1 ⊗ eΛa +ξ ) ⊗ eQν+η .
(B.1)
b N )Theorem B.7. [6] The spaces Vb (Λa +ν, ν) (a = 0, · · · , N ) give the level-1 irreducible Uq,p (sl modules with the highest weight (Λa + ν, ν), where the highest weight vectors are given by 1 ⊗ eΛa ⊗ eQν¯ . The action of the elliptic currents is given by ∑ 1 Pαj −1 Ej (z) = : exp − αj,n z −n : eαbj e−Qαj z hαj +1 (q N −j z)− r∗ , [n]q n̸=0 ∑ 1 (P +h)αj −1 ′ r Fj (z) = : exp αj,n z −n : e−bαj z −hαj +1 (q N −j z) , [n]q
(B.2)
(B.3)
n̸=0
(1 ≤ j ≤ N − 1) together with Hj± (z), Kj+ (z) in Sec.A.4 and N N 1 ∑ 1 ∑ db = d + ∗ ((P + h)j + 2)(P + h)j , (Pj + 2)P j − 2r 2r j=1
j=1
d=−
N −1 N −1 ∑ ∑ m2 1 − p∗m m 1∑ hj hj − q αj,−m Ajm . 2 [m] 1 − pm j=1
′ In (B.3) we set αj,n =
C
j=1 m∈Z>0
1 − p∗n n q αj,n . 1 − pn
Proof of Proposition 5.10
Let λ, λ′ ∈ NN , |λ| = m, |λ′ | = n and consider I = Iµ1 ···µm = (I1 , · · · , IN ) ∈ Iλ and I ′ = ′ ) ∈ I ′ . For each I ′ = {i′ , · · · , i′ } (l = 1, · · · , N ), let us set Ie′ = Iµ′1 ···µ′m = (I1′ , · · · , IN l λ l,1 l l,λ′ l
39
{m + i′l,1 , · · · , m + i′l,λ′ }. Then define I + I ′ = ((I + I ′ )1 , · · · , (I + I ′ )N ) ∈ Iλ+λ′ by (I + I ′ )l = l Il ∪ Ie′ l (l = 1 · · · , N ). Let us consider the m- and n-point functions ϕµ1 ···µm (z1 , · · · , zm ) and ϕµ′1 ,··· ,µ′n (z1′ , · · · , zn′ ). Their composition gives the m + n-point function ϕµ1 ,··· ,µm+n (z1 , · · · , zm+n ) = ϕµ1 ···µm (z1 , · · · , zm )ϕµ′1 ,··· ,µ′n (z1′ , · · · , zn′ ),
(C.1)
where we set zm+k := zk′ and µm+k := µ′k (k = 1, · · · , n). On the otherhand, from Theorem 4.2 we have I e z)ωµ ···µm (t, z, ΠI ), ϕµ1 ,··· ,µm (z1 , · · · , zm ) = dt Φ(t, 1 TM I e ′ , z ′ )ωµ′ ···µ′ (t′ , z ′ , Π′ ′ ), ϕµ′1 ,··· ,µ′n (z1′ , · · · , zn′ ) = dt′ Φ(t I n 1 ′ M T I ′ e e t˜, z ∪ z ′ )ωµ ···µ ˜ ϕµ1 ,··· ,µm+n (z1 , · · · , zm+n ) = dt˜ Φ( m+n (t, z ∪ z , Π). 1
(C.2) (C.3) (C.4)
TM +M ′
Here we set z = (z1 , · · · , zm ), z ′ = (z1′ , · · · , zn′ ), M = (1)
(N −1)
(1)
λ
+λ
′ (1)
(N −1)
j=1
(N − j)λj , M ′ = ′ (1)
′ (N −1)
∑N −1 j=1
(N − j)λ′j ,
′ (N −1)
, · · · , tλ(N −1) ), t′ = (t1 , · · · , tλ′ (1) , · · · , t1 , · · · , tλ′ (N −1) ) and (N −1) (N −1) e = {Π e µ ,j (k = 1, · · · , m + n, j = , · · · , t˜1 , · · · , t˜ (N −1) ′ (N −1) ) and Π k ′ (1)
t = (t1 , · · · , tλ(1) , · · · , t1 (1) (1) t˜ = (t˜1 , · · · , t˜ (1)
∑N −1
λ
+λ
µk + 1, · · · , N )}. Substituting (C.2)-(C.4) into (C.1), one can obtain a relation among the weight functions e ωµ1 ···µm (t, z, ΠI ), ωµ′1 ···µ′n (t′ , z ′ , Π′I ′ ) and ωµ1 ···µm+n (t˜, z ∪ z ′ , Π). Definition C.1. For the weight functions ωµ1 ···µm (t, z, Π) and ωµ′1 ···µ′n (t′ , z ′ , Π′ ), we define the ⋆-product as follows. (ωµ1 ···µm ⋆ ωµ′1 ···µ′n )(t ∪ t′ , z ∪ z ′ , ΠI+I ′ ) [ ] ∑ −2 n ϵµ′ ,h> 1 j=1 Sym
1≤kSym
k=1 l=1
∏ 1≤aSym ′ (l)
′ (l)
λ λ∏ ∏
Sym < Fl (t(l) . a )Fl (tb ) >
a=1 b=1
e t˜, z ∪ z ′ ) under the identification (C.5). Then it turns out that Υ(t, t′ , z, z ′ ) coinsides with Φ( Hence we have ϕµ1 ···µm+n (z1 , · · · , zm+n ) I ∑ ϵµ′ ,h> −2 n j=1 1 j=1
